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Drawing disconnected graphs on the Klein bottle
Laurent Beaudou, Antoine Gerbaud, Roland Grappe, Frédéric Palesi
To cite this version:
Laurent Beaudou, Antoine Gerbaud, Roland Grappe, Frédéric Palesi. Drawing disconnected graphs on the Klein bottle. 2008. �hal-00363494�
Institut Fourier
Institut Fourier
Unit´e Mixte de Recherche 5582 CNRS – Universit´e Joseph Fourier
Drawing disconnected graphs on the Klein bottle
Laurent Beaudou1, Antoine Gerbaud1, Roland Grappe2 and Fr´ed´eric Palesi1
14 mars 2008
Drawing disconnected graphs on the Klein bottle
Laurent Beaudou1, Antoine Gerbaud1, Roland Grappe2 and Fr´ed´eric Palesi1
1Institut Fourier 100, rue des Maths
38 402 St-Martin d’H`eres – FRANCE
2Laboratoire G-Scop 46 avenue F´elix Viallet 38 000 Grenoble – FRANCE.
laurent.beaudou@ujf-grenoble.fr antoine.gerbaud@ujf-grenoble.fr roland.grappe@g-scop.inpg.fr frederic.palesi@ujf-grenoble.fr
Abstract/R´esum´e
We prove that two disjoint graphs must always be drawn separately on the Klein bottle in order to minimize the crossing number of the whole drawing.
Keywords: Klein bottle, topological graph theory, crossing number.
Dans ce rapport, nous prouvons que deux graphes disjoints doivent toujours ˆetre dessin´es s´epar´ement sur la bouteille de Klein lorsque le nombre de croisements du dessin est minimal.
Mots-cl´es: bouteille de Klein, graphes topologiques, croisements.
Introduction
All graphs in this paper are finite, undirected and without loops. Apath ofG is a sequence of vertices v0, . . . , vk of Gsuch that for each integer ibetween 1 and k−1, vivi+1 is an edge of Gand all edges are distinct. Acircuit ofG is a pathv0, . . . , vk such that v0=vk. A graph with a circuit that visits each of its edges exactly once is called eulerian. A graph isconnected if for every pair of vertices uandv there is a pathv0, . . . , vk such that v0=uandvk =v. We refer to [1] for an introduction to graph theory.
Asurfaceis a two-dimensional manifold, with or without boundary. Accord- ing to [2], there are two infinite classes of compact connected surfaces without boundary: the orientable surfaces homeomorphic to a sphere with handles at- tached, and the non-orientable surfaces homeomorphic to a connected sum of projective planes. For an orientable surface, the number of handles is called the orientable genus. For a non-orientable surface, the number of projective planes is called the non-orientable genus. The non-orientable surfaces of genus 1 and 2 are the projective plane and the Klein bottle, respectively. Formal definitions of these surfaces can be found in [13].
Every curve considered throughout this paper is undirected and we do not distinguish between a curve and its image. Adrawingof a graphGon a surface Σ is a representation Ψ of Gon Σ where vertices are distinct points of Σ, and edges are curves of Σ joining the points corresponding to their endvertices. A drawing isproperif edges are simple curves without vertices of the graph in their interiors. A crossing is a transversal intersection of two curves on Σ. In this paper, we restrict our attention to proper drawings where two incident edges do not cross each other, two non-incident edges cross at most once and no more than two edges cross at a single point. The crossing number of a drawing Ψ, denoted by cr(Ψ), is the number of crossings between each pair of curves in Ψ. The crossing number of a graphGon a surface Σ is the minimum crossing number among all drawing of G on Σ. A drawing that achieves the crossing number of a graph is said optimal. By definition, a drawing with no crossing is an embedding. For background material about topological graph theory, the reader can refer to [9].
The crossing number of a graph on a surface leads to many unsolved prob- lems, see [4, 10]. DeVos, Mohar and Samal conjectured the following in [3].
Conjecture 1. Let Gbe the disjoint union of two connected graphs H andK and let Σ be a surface. For every optimal drawing of Gon Σ, the restrictions toH andK do not intersect.
This conjecture is obviously true for the sphere or equivalently for the Eu- clidean plane. It was announced proved for the projective plane in [3]. The problem remains open in the general case. In this paper, we prove that Conjec- ture 1 holds if Σ is the Klein bottle.
Theorem 2. Let G be the disjoint union of two connected graphsH and K.
For every optimal drawing of G on the Klein bottle, the restrictions toH and K do not cross.
We introduce the following notations. A closed curve is one-sided if its neighborhood is a M¨obius strip,two-sidedotherwise. There exist two non freely homotopic one-sided simple curves a and b on the Klein bottle, a two-sided
simple curve m that cuts open the Klein bottle into a cylinder, and a two- sided simple curve ethat separates the Klein bottle into two M¨obius strips. A closed curve not contractible is calledessential. According to Kawrencenko and Negami in [6], each essential simple closed curve on the Klein bottle is freely homotopic to eithera,b,m ore.
For each curve c on Σ, [c] denotes the set of curves freely homotopic to c. For each couple of curves (c, d), cr([c],[d]) denotes the minimum number of crossings, counting multiplicities, taken over all couples of [c]×[d]. Letc be a curve on Σ andI a collection of curves. The number of crossings betweencand I is denoted by cr(c, I). The minimum of cr(c′, I) taken over all curvesc′ in [c]
is denoted by cr([c], I). If I is a drawing of a graphG, the minimum cr([c], I) is taken on the curves in [c] that do not contain any vertex ofG.
We define two relations on freely homotopy classes of closed curves on the Klein bottle. Two classes [c] and [d] are said to be orthogonal if cr([c],[d])≥1, otherwise disjoint. These definitions slightly differ from those of Luo in [8]. Let Ψ be a drawing on the Klein bottle. The circuitsc of Ψ orthogonal to [a] and disjoint from [b] are calleda-circuits. The circuits orthogonal to [b] and disjoint from [a] are called b-circuits. The circuits orthogonal to [a] and [b] are called m-circuits. Finally, the circuits orthogonal to [m] and disjoint from [a] and [b]
are callede-circuits.
We will apply the following result.
Theorem 3. (De Graaf, Schrijver [5]) Let Ψ be an embedding of an eulerian graph on a metrizable surface Σ. Then Ψ can be decomposed into a collection of circuits I such that for each closed curvecon Σ,
cr([c],Ψ) =X
d∈I
cr([c],[d]).
Decomposing a drawing Ψ of a graphGinto a collection of circuitsImeans that each edge of G in Ψ is visited by exactly once by one single element of I. A related result has been obtained by Lins [7] for the projective plane and generalized by Schrijver [11] for the Klein bottle: the maximum number of pairwise edge-disjoint one-sided circuits equals the minimum number of edges intersecting all one-sided circuits. We provide another expression of this number in 7. Similarly, we express the maximum number of edge-disjointa-circuits in 8.
This paper is organized as follows. We express the maximum number of circuits of given homotopy class in section 2. Therefore we get a lower bound on the crossing number of a drawing. A case study in Section 3 guaranties that suitable transformations, defined in Section 1, yield a representation with strictly fewer crossings.
1 Drawing graphs on surfaces of smaller genus
In this section, starting with a drawing of a graph on a surface, we define new drawings of the same graph on surfaces of smaller genus. We compute the crossing numbers of these drawings.
4
1.1 Removing a crosscap
Let Σ be the non-orientable surface of genus g and Σ′ be the non-orientable surface of genusg−1.
Proposition 4. Let Ψdenote a drawing of a graphGonΣ. Letc be a simple closed one-sided curve on Σwhich does not contain any vertex of G. There is a drawingΨ′ of GonΣ′ such that
cr(Ψ′) = cr(Ψ) +cr(c,Ψ)(cr(c,Ψ)−1)
2 .
Proof. We cut open Σ alongcand we obtain the non-orientable surface Σ′ with one hole. We can glue a disk D along the boundary component to obtain Σ′. Let Ψ′ be the drawing ofGdefined by restricting Ψ to Σ\cand redrawing the edges of Ψ that crossed c onD, crossing exactly once pairwise. The crossings of these edges add to the crossings of Ψ to give the correct number of crossings of Ψ′ stated in Proposition 4.
The non-orientable surface of genusgcan be seen as a sphere withgcrosscaps attached. Attaching a crosscap to a surface Σ means removing an open diskD of Σ and identifying opposite points on the boundary of D.
Corollary 5. Let Gbe the disjoint union of two eulerian connected graphs H and K. If G has a drawing on the projective plane such that the restrictions to H and K are embeddings that cross each other, then we can find another drawing of Gon the projective plane with strictly fewer crossings such that the restrictions toH andK do not cross.
Proof. Let Ψ be a drawing ofGon the projective plane such that the restriction ΨH toH and the restriction ΨK toK are embeddings.
All one-sided simple essential closed curves on the projective plane are freely homotopic. Let c be such a curve. By a theorem of Lins [7], the maxi- mum number of edge-disjoint one-sided circuits of ΨH and ΨK are cr([c],ΨH) and cr([c],ΨK), respectively. We may assume that cr([c],ΨH) is smaller than cr([c],ΨK).
Two one-sided circuits cross at least once. Hence, each one-sided circuit of ΨH crosses each one-sided circuit of ΨK, and
cr(Ψ)≥cr([c],ΨH)×cr([c],ΨK).
Let c′ be an one-sided closed curve on the projective plane that achieves cr([c],ΨH). By Proposition 4, there exists a drawing Ψ′H ofGon the Euclidean plane such that
cr(Ψ′H) = 1
2cr(c′,ΨH)×(cr(c′,ΨH)−1)
< cr([c],ΨH)×cr([c],ΨK)
≤ cr(Ψ).
Let Ψ′denote the drawing ofGon the projective plane obtained by disjoint union of the drawings Ψ′H and ΨK. The drawings Ψ′H and ΨK do not cross each other and since ΨK is an embedding, all crossings of Ψ′ are crossings of Ψ′H. It follows that the drawing Ψ′ of Gon the projective plane has strictly fewer crossings than Ψ and the restrictions toH andK do not cross.
1.2 Removing the two crosscaps of the Klein bottle
Proposition 6. LetΨdenote a drawing of a graphGon the Klein bottle. Leta′ be a simple curve freely homotopic toaandm′ a simple curve freely homotopic tomsuch that neithera′ norm′ contains any vertex ofG, and such thata′ and m′ cross only once. Then there is a drawing Ψ′ of G on the Euclidean plane such that
cr(Ψ′) = cr(Ψ) + cr(a′,Ψ)×cr(m′,Ψ) + 1
2cr(m′,Ψ)×(cr(m′,Ψ)−1).
Proof. We cut the Klein bottle open along m′, disconnecting cr(m′,Ψ) edges of Ψ. By definition of m, the resulting surface is a cylinder. We reconnect the cut edges such that their new part remains in a small neighborhood of a′, creating exactly cr(a′,Ψ) crossings for each cut edge. Moreover, we can draw the cr(m′,Ψ) edges so that they cross each other only once. We obtain a drawing of Gon the cylinder with the desired crossing number, therefore a drawing Ψ′ on the Euclidean plane with the same crossing number.
2 Maximum number of pairwise edge-disjoint circuits
In this section we estimate the number of essential circuits of the representation.
2.1 One-sided circuits
Proposition 7. LetΨbe an embedding of an eulerian graph on the Klein bottle.
Then the maximum number of pairwise edge-disjoint one-sided circuits equals min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}.
Moreover, we can decompose Ψinto a collection of circuits I that achieves the maximum number of one-sided circuits and such that the number of m-circuits in I is
1
2(cr([a],Ψ) + cr([b],Ψ)−min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}). Proof. Let Ψ be an embedding of an eulerian graph G on the Klein bottle.
Consider a collectionI of edge-disjoint one-sided circuits of Ψ. Every one-sided circuit intersects eitheraorb. Consequently, for each circuitcinI, cr([a], c)≥1 or cr([b], c)≥1. Hence,
cr([a],Ψ) + cr([b],Ψ)≥X
c∈I
(cr([a],[c]) + cr([b],[c]))≥ |I|.
Similarly, every one-sided circuit intersects m. Hence, cr([m],Ψ)≥X
c∈I
cr([m],[c])≥ |I|.
Therefore the maximum number of pairwise edge-disjoint one-sided circuits is smaller than min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}.
6
To complete the proof of Proposition 7, it remains to decompose Ψ into a collection of circuits that contains
min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}
one-sided circuits and 1
2cr([a],Ψ) + cr([b],Ψ)−min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}
m-circuits.
LetI be a collection of circuits given by Theorem 3, withna a-circuits,nb b-circuits, nm m-circuits and ne e-circuits. By definition of I, the following equalities hold:
cr([a],Ψ) = na+nm, cr([b],Ψ) = nb+nm, cr([m],Ψ) = na+nb+ 2ne.
(1) Ifnm or ne equals zero, then the result follows. Now assume that nm and ne are positive.
Let r = min{nm, ne}. Consider r distinct m-circuits m1, . . . , mr and r distinct e-circuits e1, . . . , er in I. For every integer i between 1 and r, the circuits mi andei intersect and can be decomposed into ana-circuitai and an b-circuitbi. Thus, we getna+r a-circuits,nb+r b-circuits,nm−r m-circuits andne−r e-circuits. The resulting collection of circuitsI′ still decomposes Ψ.
By (1),
(na+r) + (nb+r) = min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}, and
2(nm−r) = cr([a],Ψ) + cr([b],Ψ)−min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}. ThusI′ is the desired collection of circuits.
2.2 a-circuits
Proposition 8. LetΨbe an embedding of an eulerian graph on the Klein bottle.
Then the maximum number of edge-disjoint a-circuits equals min{cr([a],Ψ),cr([m],Ψ)}.
Proof. Let Ψ be an embedding of an eulerian graph on the Klein bottle. Con- sider a collectionI of edge-disjointa-circuits of Ψ. Everya-circuit intersectsa.
Hence,
cr([a],Ψ)≥X
c∈I
cr([a], c)≥ |I|.
Similarly, everya-circuit intersectsm. Hence, cr([m],Ψ)≥X
c∈I
cr([m], c)≥ |I|.
Therefore the maximum number of pairwise edge-disjoint a-circuits is smaller than min{cr([a],Ψ) + cr([b],Ψ),cr([m],Ψ)}.
To complete the proof of Proposition 8, it remains to exhibit min{cr([a],Ψ),cr([m],Ψ)}
a-circuits.
Let I be a collection of circuits of Ψ as stated in Theorem 3, with na a- circuits,nb b-circuits,nmm-circuits and ne e-circuits. Ifna ornm equals zero, then the result follows. Now assume that na andnmare positive.
Let r = min{nm, ne}. Consider r distinct m-circuits m1, . . . , mr and r distinct e-circuits e1, . . . , er in I. For every integer i between 1 and r, the circuits mi andei intersect and can be decomposed into ana-circuitai and an b-circuitbi. Consequently, we get na+r a-circuits, nb+r b-circuits, nm−r m-circuits and ne −r e-circuits. The resulting collection of circuits I′ still decomposes Ψ.
Lets= min{nm−r, nb+r}. Ifnm ≤ne, thens equals zero and we have found cr([a],Ψ) a-circuits. Otherwise, consider s distinct b-circuits b1, . . . , bs and s distinct m-circuits m′1, . . . , m′s ofI′. For every integer i between 1 and r, the circuits bi and m′i intersect and can be decomposed into ana-circuita′i. And so, we getna+r+s a-circuits.
By (1),
na+r+m = na+r+ min{nm−r, nb+r}
= na+ min{nm, nb+ 2ne}
= min{cr([a],Ψ),cr([m],Ψ)}. Proposition 8 is proved.
Note that Proposition 8 still holds when replacingabyb.
3 Main result
This section is devoted to the proof of our main result. First, we need to prove the following special case of the theorem.
Lemma 9. LetGbe the disjoint union of two eulerian connected graphsH and K. If Ghas a drawing on the Klein bottle such that the restrictions to H and K are embeddings that cross each other, then we can find another drawing ofG on the Klein bottle with strictly fewer crossings such that the restrictions to H andK do not cross.
Proof. Let Ψ be a drawing ofG on the Klein bottle such that the restrictions ΨH toH and ΨK toK are embeddings.
To prove Lemma 9, it is enough to find two drawings Ψ′H and Ψ′K of H andK on two disjoint subsurfaces of the Klein bottle such that the sum of the crossings of Ψ′H and the crossings of Ψ′K is strictly less than the crossings of Ψ.
Indeed, let Ψ′H and Ψ′K be such drawings and let Ψ′ denote the drawing of G on the Klein bottle plane obtained by disjoint union of the drawings Ψ′H and Ψ′K. Then the number of crossings of Ψ′ is the sum of the crossings of Ψ′H and the crossings of Ψ′K. It follows that the drawing Ψ′ofGon the Klein bottle has strictly fewer crossings than Ψ and the restrictions toH andKdo not cross.
For convenience, we denote cr([a],ΨH), cr([b],ΨH) and cr([m],ΨH) byha,hb andhm, respectively. Similarly, we denote cr([a],ΨK), cr([b],ΨK) and cr([m],ΨK) byka,kb andkm, respectively.
8
Assume without loss of generality thathm≤km.
By Proposition 7, there exist a decomposition of ΨH into pairwise edge-
disjoint circuits with min{hm, ha+hb}one-sided circuits and (ha+hb−min{hm, ha+hb})/2 m-circuits. Each one-sided circuit crosses ΨK at least min{ka, kb} times, and
each m-circuit crosses ΨK at least km times. Counting the crossings between ΨH and ΨK gives the following inequality.
cr(Ψ)≥min{hm, ha+hb}×min{ka, kb}+1
2(ha+hb−min{hm, ha+hb})×km. (2) With a similar decomposition of ΨK we obtain
cr(Ψ)≥min{km, ka+kb}×min{ha, hb}+1
2(ka+kb−min{km, ka+kb})×hm. (3) Beside, by Proposition 8, there exist min{ka, km} pairwise edge-disjointa- circuits of ΨK. Each of them crosses ΨH at leastha times, therefore
cr(Ψ)≥min{ka, km} ×ha. (4) Similarly, consideringb-circuits gives
cr(Ψ)≥min{kb, km} ×hb. (5) Letm1≤m2≤m3≤m4 be an ordering of the numbersha, hb, ka, kb.
(Case 1) If km ≥ m2, then applying twice Proposition 4 provides a drawing Ψ′H of H and a drawing Ψ′K of K on disjoint subsurfaces of the Klein bottle such that
cr(Ψ′H) + cr(Ψ′K) =1
2m1×(m1−1) +1
2m2×(m2−1).
By definition ofm2 and sincekm≥m2,
m2×m2≤max (min{ka, km} ×ha,min{kb, km} ×hb). Hence, by (4) and (5),
cr(Ψ′H) + cr(Ψ′K)< m2×m2≤cr(Ψ).
(Case 2)Ifkm< m2, then
hm≤km≤m1+m2≤ha+hb. Thus, (2) becomes
cr(Ψ)≥hm×min{ka, kb}+1
2(ha+hb−hm)×km. (6) Sincekm≤ka+kb, the (3) becomes
cr(Ψ)≥km×ha+1
2(ka+kb−km)×hm. (7)
(Case 2.1)Ifhm≤ka+kb−km, then by Proposition 6 there exists a drawing Ψ′H ofH on the Euclidean plane such that
cr(Ψ′H) =hm×ha+1
2hm×(hm−1).
We get by (7)
cr(Ψ′H) + cr(ΨK) =hm×ha+1
2hm×(hm−1)
≤km×ha+1
2(ka+kb−km)×(hm−1)
<cr(Ψ).
(Case 2.2) Ifhm> ka+kb−km, thenkm< m2 implies hm+ max{ka, kb} ≥hm+km> ka+kb. Hence hm>min{ka, kb}=m1.
(Case 2.2.1)Ifm1< km/2 then we apply Proposition 6. There exists a drawing Ψ′K ofK on the Euclidean plane such that
cr(Ψ′K) =km×m1+1
2km×(km−1).
By (6),
cr(ΨH) + cr(Ψ′K) ≤ km×m1+1
2km×(km−1)
≤ hm×m1+ (km−hm)×m1+1
2km×(km−1)
< hm×m1+ (2km−hm)×1 2km
< hm×min{ka, kb}+1
2(ha+hb−hm)×km
< cr(Ψ).
(Case 2.2.2)Ifm1≥km/2 andm2<2km, then applying twice Proposition 4 provides a drawing Ψ′H ofH and a drawing Ψ′K ofK on disjoint subsurfaces of the Klein bottle such that
cr(Ψ′H) + cr(Ψ′K) =1
2m1×(m1−1) +1
2m2×(m2−1).
Since
m1< hm, m2> km, m2≤ha and m1+m2≤ka+kb, we get, by (7),
cr(Ψ′H) + cr(Ψ′K) = 1
2m2×(m2−1) +1
2m1×(m1−1)
< 1
2m2×2km+1
2m1×(hm−1)
< ha×km+1
2(m1+m2−km)×hm
< ha×km+1
2(ka+kb−km)×hm
< cr(Ψ).
10
(Case 2.2.3)Ifm1≥km/2 andm2≥2kmthen we apply Proposition 6. There exists a drawing Ψ′K ofK on the Euclidean plane such that
cr(Ψ′K) =km×m1+1
2km×(km−1).
Hence, by (4),
cr(ΨH) + cr(Ψ′K) = km×m1+1
2km×(km−1)
< km×
m1+1 2km
< km×(2m1)
< m1×m2
< cr(Ψ).
Now, we may prove the Theorem.
Theorem 10. Let G be the disjoint union of two connected graphs H andK.
For every optimal drawing of G on the Klein bottle, the restrictions toH and K do not cross.
Proof. LetG be the disjoint union of two connected graphsH and K. Let Ψ be an optimal drawing ofGon the Klein bottle.
First, assume that the restrictions toH and K are embeddings. Duplicate each edge of Gand denote byG′, H′,K′ the resulting eulerian graphs and by Ψ′ the resulting drawing. The drawing Ψ′ has 4 cr(Ψ) crossings. Since G′ is eulerian, by Lemma 9, we can find a drawing Ψ′′ of G′, where the restrictions to H′ and K′ do not cross each other, with strictly less than 4 cr(Ψ) crossings.
Moreover, we can assume that every pair of parallel edges are drawn close enough to have the same crossings. Therefore, two pairs of parallel edges have either four crossings or none. Thus, by deleting one copy of each edge, we get a drawing of Gwith strictly less than cr(Ψ) crossings. Moreover, the drawings of H and K do not intersect.
Secondly, suppose that the restrictions to H and K are not embeddings.
Consider the graphs H′′ and K′′ obtained from H and K by adding a vertex for each internal crossing. The corresponding drawings are embeddings and we apply what was shown just above. Theorem 2 is proved when we replace the new vertices by the former crossings.
References
[1] B. Bollob´as, Modern Graph Theory,Springer, New-York, 1998
[2] H. R. Brahana, Systems of circuits of two-dimensional manifolds,Ann. of Math.,30(1923), 234–243
[3] M. DeVos, B. Mohar and R. Samal,Open Problems Garden,
http://garden.irmacs.sfu.ca/?q=op/drawing disconnected graphs on surfaces, 2007
[4] P. Erd¨os and R. K. Guy, Crossing number problems,The American Math- ematical Monthly,80(1973), 52–57
[5] M. de Graaf and A. Schrijver, Decomposition on surfaces,J. Combin. The- ory Ser. B, 70(1997), 157–165
[6] S. Kawrencenko and S. Negami, Irreducible triangulations of the Klein Bottle, J. Combin. Theory Ser. B, 70 (1997), 265–291, doi:10.1006/jctb.1997.9999
[7] S. Lins, A minimax theorem on circuits in projective graphs,J. Combin.
Theory Ser. B,30(1981), 253–262, doi:10.1016/0095-8956(81)90042-3 [8] F. Luo, On non-separating simple closed curves in a compact surface,Topol-
ogy,36(2)(1997), 381–410
[9] B. Mohar and C. Thomassen, Graphs on surfaces,Johns Hopkins University Press, Baltimore, 2001
[10] J. Pach and G. T´oth, Thirteen problems on crossing numbers,Geombina- torics, 9(2000), 195–207
[11] A. Schrijver, The Klein bottle and multicommodity flows, Combinatorica, 9(1989), 375–384, doi:10.1007/BF02125349
[12] A. Schrijver, Decomposition of graphs on surfaces and a homotopic circu- lation theorem,J. Combin. Theory Ser. B,51(1991), 161–210
[13] J. Stillwell, Classical topology and combinatorial group theory, Springer- Verlag, New York, 1980, 62–67
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